Cauchy Combination Test: A Powerful Test With Analytic p-Value Calculation Under Arbitrary Dependency Structures

Abstract-Combining individual p-values to aggregate multiple small effects has a long-standing interest in statistics, dating back to the classic Fisher's combination test. In modern large-scale data analysis, correlation and sparsity are common features and efficient computation is a necessary...

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Published in:	Journal of the American Statistical Association Vol. 115; no. 529; pp. 393 - 402
Main Authors:	Liu, Yaowu, Xie, Jun
Format:	Journal Article
Language:	English
Published:	United States Taylor & Francis 02.01.2020 Taylor & Francis Ltd
Subjects:	Arbitrariness Cauchy distribution Computation Computer simulation Correlation analysis Correlation matrix Data analysis Dependence Dependency Dependency grammar Distribution Genomics Global hypothesis testing High-dimensional data Mathematical analysis Nonasymptotic approximation Power Regression analysis Sparse alternative Sparsity Statistical methods Statistical tests Statistics Transformation Values Global hypothesis testing Sparse alternative Non-asymptotic approximation High dimensional data Cauchy distribution Correlation matrix
ISSN:	0162-1459, 1537-274X
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Abstract	Abstract-Combining individual p-values to aggregate multiple small effects has a long-standing interest in statistics, dating back to the classic Fisher's combination test. In modern large-scale data analysis, correlation and sparsity are common features and efficient computation is a necessary requirement for dealing with massive data. To overcome these challenges, we propose a new test that takes advantage of the Cauchy distribution. Our test statistic has a simple form and is defined as a weighted sum of Cauchy transformation of individual p-values. We prove a nonasymptotic result that the tail of the null distribution of our proposed test statistic can be well approximated by a Cauchy distribution under arbitrary dependency structures. Based on this theoretical result, the p-value calculation of our proposed test is not only accurate, but also as simple as the classic z-test or t-test, making our test well suited for analyzing massive data. We further show that the power of the proposed test is asymptotically optimal in a strong sparsity setting. Extensive simulations demonstrate that the proposed test has both strong power against sparse alternatives and a good accuracy with respect to p-value calculations, especially for very small p-values. The proposed test has also been applied to a genome-wide association study of Crohn's disease and compared with several existing tests. Supplementary materials for this article are available online.
AbstractList	Combining individual -values to aggregate multiple small effects has a long-standing interest in statistics, dating back to the classic Fisher's combination test. In modern large-scale data analysis, correlation and sparsity are common features and efficient computation is a necessary requirement for dealing with massive data. To overcome these challenges, we propose a new test that takes advantage of the Cauchy distribution. Our test statistic has a simple form and is defined as a weighted sum of Cauchy transformation of individual -values. We prove a non-asymptotic result that the tail of the null distribution of our proposed test statistic can be well approximated by a Cauchy distribution under arbitrary dependency structures. Based on this theoretical result, the -value calculation of our proposed test is not only accurate, but also as simple as the classic -test or -test, making our test well suited for analyzing massive data. We further show that the power of the proposed test is asymptotically optimal in a strong sparsity setting. Extensive simulations demonstrate that the proposed test has both strong power against sparse alternatives and a good accuracy with respect to -value calculations, especially for very small -values. The proposed test has also been applied to a genome-wide association study of Crohn's disease and compared with several existing tests. Abstract-Combining individual p-values to aggregate multiple small effects has a long-standing interest in statistics, dating back to the classic Fisher's combination test. In modern large-scale data analysis, correlation and sparsity are common features and efficient computation is a necessary requirement for dealing with massive data. To overcome these challenges, we propose a new test that takes advantage of the Cauchy distribution. Our test statistic has a simple form and is defined as a weighted sum of Cauchy transformation of individual p-values. We prove a nonasymptotic result that the tail of the null distribution of our proposed test statistic can be well approximated by a Cauchy distribution under arbitrary dependency structures. Based on this theoretical result, the p-value calculation of our proposed test is not only accurate, but also as simple as the classic z-test or t-test, making our test well suited for analyzing massive data. We further show that the power of the proposed test is asymptotically optimal in a strong sparsity setting. Extensive simulations demonstrate that the proposed test has both strong power against sparse alternatives and a good accuracy with respect to p-value calculations, especially for very small p-values. The proposed test has also been applied to a genome-wide association study of Crohn's disease and compared with several existing tests. Supplementary materials for this article are available online. Combining individual p-values to aggregate multiple small effects has a long-standing interest in statistics, dating back to the classic Fisher's combination test. In modern large-scale data analysis, correlation and sparsity are common features and efficient computation is a necessary requirement for dealing with massive data. To overcome these challenges, we propose a new test that takes advantage of the Cauchy distribution. Our test statistic has a simple form and is defined as a weighted sum of Cauchy transformation of individual p-values. We prove a non-asymptotic result that the tail of the null distribution of our proposed test statistic can be well approximated by a Cauchy distribution under arbitrary dependency structures. Based on this theoretical result, the p-value calculation of our proposed test is not only accurate, but also as simple as the classic z-test or t-test, making our test well suited for analyzing massive data. We further show that the power of the proposed test is asymptotically optimal in a strong sparsity setting. Extensive simulations demonstrate that the proposed test has both strong power against sparse alternatives and a good accuracy with respect to p-value calculations, especially for very small p-values. The proposed test has also been applied to a genome-wide association study of Crohn's disease and compared with several existing tests.Combining individual p-values to aggregate multiple small effects has a long-standing interest in statistics, dating back to the classic Fisher's combination test. In modern large-scale data analysis, correlation and sparsity are common features and efficient computation is a necessary requirement for dealing with massive data. To overcome these challenges, we propose a new test that takes advantage of the Cauchy distribution. Our test statistic has a simple form and is defined as a weighted sum of Cauchy transformation of individual p-values. We prove a non-asymptotic result that the tail of the null distribution of our proposed test statistic can be well approximated by a Cauchy distribution under arbitrary dependency structures. Based on this theoretical result, the p-value calculation of our proposed test is not only accurate, but also as simple as the classic z-test or t-test, making our test well suited for analyzing massive data. We further show that the power of the proposed test is asymptotically optimal in a strong sparsity setting. Extensive simulations demonstrate that the proposed test has both strong power against sparse alternatives and a good accuracy with respect to p-value calculations, especially for very small p-values. The proposed test has also been applied to a genome-wide association study of Crohn's disease and compared with several existing tests. Abstract–Combining individual p-values to aggregate multiple small effects has a long-standing interest in statistics, dating back to the classic Fisher’s combination test. In modern large-scale data analysis, correlation and sparsity are common features and efficient computation is a necessary requirement for dealing with massive data. To overcome these challenges, we propose a new test that takes advantage of the Cauchy distribution. Our test statistic has a simple form and is defined as a weighted sum of Cauchy transformation of individual p-values. We prove a nonasymptotic result that the tail of the null distribution of our proposed test statistic can be well approximated by a Cauchy distribution under arbitrary dependency structures. Based on this theoretical result, the p-value calculation of our proposed test is not only accurate, but also as simple as the classic z-test or t-test, making our test well suited for analyzing massive data. We further show that the power of the proposed test is asymptotically optimal in a strong sparsity setting. Extensive simulations demonstrate that the proposed test has both strong power against sparse alternatives and a good accuracy with respect to p-value calculations, especially for very small p-values. The proposed test has also been applied to a genome-wide association study of Crohn’s disease and compared with several existing tests. Supplementary materials for this article are available online. Combining individual p-values to aggregate multiple small effects has a long-standing interest in statistics, dating back to the classic Fisher’s combination test. In modern large-scale data analysis, correlation and sparsity are common features and efficient computation is a necessary requirement for dealing with massive data. To overcome these challenges, we propose a new test that takes advantage of the Cauchy distribution. Our test statistic has a simple form and is defined as a weighted sum of Cauchy transformation of individual p-values. We prove a non-asymptotic result that the tail of the null distribution of our proposed test statistic can be well approximated by a Cauchy distribution under arbitrary dependency structures. Based on this theoretical result, the p-value calculation of our proposed test is not only accurate, but also as simple as the classic z-test or t-test, making our test well suited for analyzing massive data. We further show that the power of the proposed test is asymptotically optimal in a strong sparsity setting. Extensive simulations demonstrate that the proposed test has both strong power against sparse alternatives and a good accuracy with respect to p-value calculations, especially for very small p-values. The proposed test has also been applied to a genome-wide association study of Crohn’s disease and compared with several existing tests.
Author	Liu, Yaowu Xie, Jun
Author_xml	– sequence: 1 givenname: Yaowu surname: Liu fullname: Liu, Yaowu organization: Department of Biostatistics, Harvard School of Public Health – sequence: 2 givenname: Jun surname: Xie fullname: Xie, Jun email: junxie@purdue.edu organization: Department of Statistics, Purdue University
BackLink	https://www.ncbi.nlm.nih.gov/pubmed/33012899$$D View this record in MEDLINE/PubMed
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Snippet	Abstract-Combining individual p-values to aggregate multiple small effects has a long-standing interest in statistics, dating back to the classic Fisher's... Combining individual -values to aggregate multiple small effects has a long-standing interest in statistics, dating back to the classic Fisher's combination... Abstract–Combining individual p-values to aggregate multiple small effects has a long-standing interest in statistics, dating back to the classic Fisher’s... Combining individual p-values to aggregate multiple small effects has a long-standing interest in statistics, dating back to the classic Fisher's combination... Combining individual p-values to aggregate multiple small effects has a long-standing interest in statistics, dating back to the classic Fisher’s combination...
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SubjectTerms	Arbitrariness Cauchy distribution Computation Computer simulation Correlation analysis Correlation matrix Data analysis Dependence Dependency Dependency grammar Distribution Genomics Global hypothesis testing High-dimensional data Mathematical analysis Nonasymptotic approximation Power Regression analysis Sparse alternative Sparsity Statistical methods Statistical tests Statistics Transformation Values
Title	Cauchy Combination Test: A Powerful Test With Analytic p-Value Calculation Under Arbitrary Dependency Structures
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